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<style|generic>

<\body>
  1. What is z in frequency domain?

  <\indent>
    z = <math|e<rsup|j w>>
  </indent>

  2. What is <math|z<rsup|-2>> in frequency domain?

  <\indent>
    <math|z<rsup|-2> = e<rsup|-2 j w>>
  </indent>

  3. Give me the simplest low pass FIR filter?

  <\indent>
    <math|<frac|1|2><around*|(|1+z<rsup|-1>|)>>
  </indent>

  4. Give me the simplest high pass FIR filter?

  <\indent>
    <math|<frac|1|2><around*|(|1-z<rsup|-1>|)>>
  </indent>

  5. What is the group delay of an FIR filter?

  <\indent>
    The middle of the FIR filter
  </indent>

  6. What is the z transform for <math|x<around*|(|n-2|)>>?

  <\indent>
    <math|X<around*|(|z|)>z<rsup|-2>>
  </indent>

  7. What is the Fourier transform for <math|x<around*|(|n-3|)>>?

  <\indent>
    <math|X<around*|(|e<rsup|j w>|)>e<rsup|-3 j w>>
  </indent>

  8. What is the z transform of <math|\<delta\><around*|[|n-4|]>>?

  <\indent>
    <math|z<rsup|-4>>
  </indent>

  9. What is the frequency response for a window low pass filter?

  <\indent>
    <math|e<rsup|-j\<alpha\>\<omega\>>> <math|\<alpha\>> is the group delay
  </indent>

  10. What is the impulse response of a window function?

  <\indent>
    <math|<frac|sin<around*|(|w<rsub|c><around*|(|n-\<alpha\>|)>|)>|\<pi\><around*|(|n-\<alpha\>|)>>>
  </indent>

  <\equation*>
    h<around*|[|n|]>=<frac|1|2\<pi\>><big|int><rsup|\<omega\><rsub|c>><rsub|-\<omega\><rsub|c>>e<rsup|-j\<alpha\>\<omega\>>e<rsup|j\<omega\>n>d\<omega\>
  </equation*>

  <\equation*>
    h<around*|[|n|]>=<frac|1|2\<pi\>><big|int><rsup|\<omega\><rsub|c>><rsub|-\<omega\><rsub|c>>e<rsup|j<around*|(|n-\<alpha\>|)>\<omega\>>d\<omega\>
  </equation*>

  <\equation*>
    h<around*|[|n|]>=<frac|1|j2\<pi\><around*|(|n-\<alpha\>|)>>e<rsup|j<around*|(|n-\<alpha\>|)>\<omega\>><around*|\|||\<nobracket\>><rsup|\<omega\><rsub|c>><rsub|-\<omega\><rsub|c>>
  </equation*>

  <\equation*>
    h<around*|[|n|]>=<frac|1|j<around*|(|n-\<alpha\>|)>><around*|[|e<rsup|j<around*|(|n-\<alpha\>|)>\<omega\><rsub|c>>-e<rsup|-j<around*|(|n-\<alpha\>|)>\<omega\><rsub|c>>|]>
  </equation*>

  <\equation*>
    h<around*|[|n|]>=<frac|1|\<pi\><around*|(|n-\<alpha\>|)>><frac|<around*|[|e<rsup|j<around*|(|n-\<alpha\>|)>\<omega\><rsub|c>>-e<rsup|-j<around*|(|n-\<alpha\>|)>\<omega\><rsub|c>>|]>|2j>
  </equation*>

  <\equation*>
    h<around*|[|n|]>=<frac|sin<around*|[|<around*|(|n-\<alpha\>|)>w<rsub|c>|]>|\<pi\><around*|(|n-\<alpha\>|)>>
  </equation*>

  11. What is the equation for magnitude square?

  <\equation*>
    <around*|\||H<around*|(|e<rsup|j\<omega\>>|)>|\|><rsup|2> =
    H<around*|(|e<rsup|j\<omega\>>|)>H<rsup|*\<ast\>><around*|(|e<rsup|j\<omega\>>|)><rsup|*>
  </equation*>

  12. Why would you want a system where both poles and zeros are within the
  unit circle?

  <\indent>
    Having the poles within the unit circle implies that the system is causal
    and stable. If the zeros are also within the unit circle, it implies that
    the inverse of the system is also causal and stable.
  </indent>

  13. How do you call a system where both the poles and zeros are within the
  unit circle?

  <\indent>
    minimum phase system
  </indent>

  14. What is one purple of all pass filter?

  <\indent>
    All pass filters can be used to adjust the group delay or the phase of a
    system without effecting the magnitude.
  </indent>

  15. What are the 2 ways to convert continuous signal into discrete ?

  <\indent>
    Impulse invariant and binlinear tansformation
  </indent>

  16. If we take a continuous signal <math|x<around*|(|t|)>> and convert
  discretize it with <math|x<around*|(|n\<Tau\>|)>>, what happens in the
  frequency domain?

  <\equation*>
    X<around*|(|e<rsup|j\<omega\>>|)>=<frac|1|T><below|<above|<big|sum>|\<infty\>>|k=-\<infty\>>X<rsub|c><around*|[|j<around*|(|<frac|\<omega\>|T>-<frac|2\<pi\>k|T>|)>|]>
  </equation*>

  <\indent>
    Important to realize that this works for both the signal and a system. If
    we are talking about a system, we would use h instead of X.
  </indent>

  17. What is the first thing we must realize about impulse invariant method?

  <\indent>
    The entire approach starts with the assumption that we are talking about
    a specific type of equation with the following equation :
  </indent>

  <\equation*>
    h<around*|(|t<rsub|>|)>=A e<rsup|\<alpha\>t>u<around*|(|t|)>
  </equation*>

  18. With impulse invariant, what is the key laplace form that corresponds
  with the <math|h<around*|(|t|)>>?

  <\equation*>
    H<around*|(|s|)>=<frac|A|s-\<alpha\>>
  </equation*>

  19. With impulse invariant, with <math|h<around*|(|t|)>>, what is the
  equation for <math|h<around*|(|n\<Tau\>|)>>?

  <\equation*>
    h<around*|(|n\<Tau\>|)>=A e<rsup|\<alpha\>n\<Tau\>>u<around*|[|n\<Tau\>|]>
  </equation*>

  20. With impulse invariant, what is <math|H<around*|(|z|)>>?

  <\equation*>
    H<around*|(|z|)>=<frac|A|1-<frac|e<rsup|\<alpha\>T>|z>>
  </equation*>

  21. What is the final concluse regarding the impulse invariant conversion?

  <\equation*>
    <frac|A|s-\<alpha\>>\<Rightarrow\><frac|A|1-<frac|e<rsup|\<alpha\>T>|z>>
  </equation*>

  22. What is the equation to a continuous signal back from discrete time?

  <\equation*>
    x<around*|(|t|)>=<below|<above|<big|sum>|\<infty\>>|n=-\<infty\>>x<around*|[|n|]><frac|sin<around*|[|\<pi\>F<rsub|s><around*|(|t-n\<Tau\>|)>|]>|\<pi\>F<rsub|s><around*|(|t-n\<Tau\>|)>>
  </equation*>

  <\indent>
    Or\ 
  </indent>

  <\equation*>
    x<around*|(|t|)>=<below|<above|<big|sum>|\<infty\>>|n=-\<infty\>>sinc<around*|[|F<rsub|s><around*|(|t-n\<Tau\>|)>|]>
  </equation*>

  <\indent>
    The definition of sinc is\ 
  </indent>

  <\equation*>
    sinc<around*|(|x|)>=<frac|sin<around*|(|\<pi\>x|)>|\<pi\>x>
  </equation*>
</body>

<\initial>
  <\collection>
    <associate|language|american>
    <associate|page-type|letter>
  </collection>
</initial>